Prove that at any time there are two opposite points along the Equator, which have exactly the same temperature. Assume the temperature function varies continuously as you move along the Equator.

Counterargument: This is patently impossible. If there are such points on the Equator, there must also be similar points on any circle around the Earth, such as a meridian. But in that case, we'd have one point in the north hemisphere, in winter, and the other in the south, in summer; that doesn't make sense!

Suppose we test the Temp at 2 points A and A' (which are exactly opposite each other) and call the temperature T and T' repectively. Let T be the temp at the point we are looking at, and T' is the temp on the opposite side of the globe. Suppose T>T'. Now imagine having the ability to test temps at B and B', C and C' simultaneously such that points A, B, C etc are an infinite series of points along the course of the equator (or any other circle around the globe). Assuming that temperature is a continuous variable, and there are no sudden discrete jumps in temperature, we will have to eventually find a pair of opposite points P and P' where the temps are equal.

PROOF:Suppose not. Then at each point temp T>T' because if we ever found 2 'nearby' points X and Y, such that T>T' at X&X', but T'>T at Y&Y', then we could be sure of finding some point between X and Y where T=T'. So we keep checking points all around the equator and keep getting T>T' until we get 180 degrees around the globe and are back to the original points A and A'. But now we are looking at A', and the opposite point is A. Therefore T'>T. But earlier we assumed T>T' Paradox.